What is the remainder when $\left(x^4+36\right)$ is divided by $\left(x^2-8\right)$?
To find the remainder when the polynomial \(P(x) = x^4 + 36\) is divided by the divisor \(D(x) = x^2 - 8\), we use the polynomial division algorithm. According to this algorithm, for any polynomials \(P(x)\) and \(D(x)\) where \(D(x) eq 0\), there exist a unique quotient \(Q(x)\) and a remainder \(R(x)\) such that \(P(x) = D(x)Q(x) + R(x)\), where the degree of \(R(x)\) is less than the degree of \(D(x)\). Here, since the degree of \(D(x) = x^2 - 8\) is 2, the degree of the remainder \(R(x)\) must be at most 1, implying \(R(x)\) is of the form \(ax + b\).
We can use algebraic manipulation to express \(P(x) = x^4 + 36\) in a way that highlights the divisor \(x^2 - 8\). First, note that \(x^4\) can be written as \((x^2)^2\). We can rewrite this term by introducing the divisor: \((x^2)^2 = (x^2 - 8 + 8)^2\). Alternatively, a simpler approach is to recognize the difference of squares: notice that \(x^4 - 64 = (x^2)^2 - 8^2 = (x^2 - 8)(x^2 + 8)\). This shows that \(x^4 - 64\) is exactly divisible by \(x^2 - 8\). We can now rewrite \(P(x)\) as:
\[P(x) = x^4 + 36 = (x^4 - 64) + 64 + 36\]
\[P(x) = (x^4 - 64) + 100\]
\[P(x) = (x^2 - 8)(x^2 + 8) + 100\]
By comparing this to the division algorithm \(P(x) = D(x)Q(x) + R(x)\), we identify \(Q(x) = x^2 + 8\) and the remainder \(R(x) = 100\).
We can verify this result by setting \(x^2 = 8\) (the root of the divisor). Substituting \(x^2 = 8\) into \(P(x) = (x^2)^2 + 36\), we get:
\[P(x) = (8)^2 + 36 = 64 + 36 = 100\]
This value \(100\) represents the constant remainder when the polynomial is divided by \(x^2 - 8\). The remainder is therefore \(\boxed{100}\).
Polynomial division is a fundamental tool in algebra used for simplifying complex expressions and solving equations. For example, engineers and physicists use this to analyze system behavior, such as determining how a signal (represented by a polynomial) changes when it passes through a specific filter (another polynomial). By finding the remainder, they can isolate the steady-state component of the system or identify key system properties like resonance, which are essential for designing stable and efficient circuits or mechanical systems.